BIO 356 Lab 8: Competition - Part 1

In this lab you will learn

• Competition arises as consumers share resources.

• Ecologists derive coexistence rules from the Lotka-Volterra competition model.

• Two species cannot stably coexist on a single resource.

• Two species can coexist when each has a stronger competitive impact on itself than on the other.

• Priority effects occur when species regulate each other more strongly than themselves.

• Species may coexist by specializing on different resources.

library(tidyverse)
library(gridExtra) ## for plotting function grid.arrange()
library(deSolve) ## for predator-prey-resource ODE model
library(magrittr) ## for pipe symbol %<>%
library(knitr) ## for table-viewing function kable()

theme_set(theme_bw())
theme_update(
  panel.grid = element_blank(),
  aspect.ratio = 1
)

Competition, along with predation/parasitism and mutualisms, is one of the most important species interactions in nature. In fact, the idea that competition for finite resources plays a key role in the living world was very much on the minds of Charles Darwin, Alfred Russell Wallace, and other early proponents of evolution by natural selection. Darwin had the insight that competition was the driving force behind adaptation.

As natural selection acts by competition, it adapts the inhabitants of each country only in relation to the degree of perfection of their associates; so that we need feel no surprise at the inhabitants of any one country, although on the ordinary view supposed to have been specially created and adapted for that country, being beaten and supplanted by the naturalised productions from another land. Charles Darwin, The Origin of Species

Figure: Galápagos finches, closely related birds in the tanager family whose diversity in beak shape influenced Darwin’s ideas about adaptation via specialization to different diets. “Seeing this gradation and diversity of structure in one small, intimately related group of birds, one might really fancy that from an original paucity of birds in this archipelago, one species had been taken and modified for different ends.” Species depicted, clockwise from top left: G. magnirostris, G. fortis, C. olivacea, C. parvulus. Photo collage by Kiwi Rex

In this lab we will see how competition arises from consumer-resource dynamics, and will use the classic Lotka-Volterra equations to refine our intuitions about when consumers sharing resources will be able to coexist.

The Lotka-Volterra competition equations

Competition has been long known to naturalists as an important driver of ecological (and ultimately evolutionary) dynamics. However, it wasn’t until the 1920s that it received its first mathematical treatment by Alfred Lotka and Vito Volterra. The so-called Lotka-Volterra model is arguably the simplest model of competition one can write. Combined with the competitive exclusion principle proposed by Georgy Gause based on his experiments with yeast and Paramecium, and later the theoretical work of Robert MacArthur linking the Lotka-Volterra equations to consumer-resource dynamics, the model sits at the core of our quantitative understanding of species coexistence.

Figure: Vito Volterra (1860-1940), Alfred Lotka (1880-1949), Georgy Gause (1910-1986), and Robert MacArthur (1930-1972), pioneers of quantitative community ecology.

Consider a system with two species which rely on the same resource. We assume the resource is biotic, and as such grows logistically in the absence of the consumers. Consumers deplete the resource in proportion to their own abundance, with a proportionality constant reflecting the degree to which they attack the resource. Consumer populations grow in proportion to the amount of resource consumed, and decline due to intrinsic mortality.

Mathematically, we can write

\[ \frac{dR}{dt}=cR\left[\left(1-\frac{R}{K}\right) - a_1N_1-a_2N_2\right] \tag{1} \]

\[ \begin{eqnarray} \frac{dN_1}{dt} = (b_1R-d_1)N_1\\ \tag{2} \frac{dN_2}{dt} = (b_2R-d_2)N_2 \end{eqnarray} \]

which you may recognize as a predator-prey system with two predators and one prey. The constants are interpreted as follows:

Q1. Recall that equilibrium is defined as the stage where none of the population sizes are changing, which mathematically corresponds to all time-derivatives being equal to zero. If there are no consumers in the ecosystem, what will be the equilibrium population of the resource?

As we have seen in the predator-prey dynamics module, resource and consumer populations will be constantly changing in response to each other. Suppose that the prey/resource has much faster dynamics than the consumer/predator. This approximation is reasonable in cases where the consumer/predator is much larger and longer-lived than the resource/prey: whales and krill, owls and mice, or krill and phytoplankton, where one generation of the predator/consumer corresponds to many generations of the prey/resource. Notice this is the opposite situation of infectious disease dynamics, where it is the predator that has faster dynamics.

Figure: The world’s largest animal, the blue whale, Balaenoptera musculus, feeds on krill, small crustaceans whose lifespans range from 6 months to 6 years. By comparison, the blue whale lives for up to 90 years. Krill in turn feed primarily on unicellular algae whose life cycle is about 6 days. Whale photo by Smithsonian Magazine. Krill photo by Auscape International Pty Ltd.

In practice, this means the resource population reaches equilibrium much faster than the consumer, because the resource population will quickly respond to changes in the consumer populations and restore dynamic equilibrium. In other words, we can assume that from the consumer perspective, it is always the case that \(\frac{dR}{dt} = 0\).

Q2. Assuming equilibrium in the resource equation, i.e. \(\frac{dR}{dt}=0\) in Equation (1), and assuming resource abundance \(R\) is not zero, write \(R\) in terms of the population sizes of the consumers, \(N_1\), \(N_2\). (You can do pen-and-paper calculations or just plug the equation in WolframAlpha.) How does this equilibrium abundance of the resource compare to its equilibrium abundance in the absence of the consumers?

Since we know how to write the resource population in terms of the consumer populations, we can replace \(R\) in Equation (2) and be left with only two unknown variables, namely the consumer populations.

Replacing the expression for \(R\) you found in Q2 in Equation (2), we get:

\[ \begin{eqnarray} \frac{dN_1}{dt} = r_1N_1\left(1 - \frac{N_1 + \alpha_{12}N_2}{K_1} \right) \\ \tag{3} \frac{dN_2}{dt} = r_2N_2\left(1 - \frac{N_2 + \alpha_{21}N_1}{K_2} \right) \end{eqnarray} \] Where we introduced a bunch of new constants,

\[ \begin{eqnarray} r_1 = b_1 K - d_1 & \hspace{1cm} & K_1 = \frac{r_1}{b_1a_1K} & \hspace{1cm} \alpha_{12} = \frac{a_2}{a_1} \\ r_2 = b_2 K - d_2 & \hspace{1cm} & K_2 = \frac{r_2}{b_2a_2K} & \hspace{1cm} \alpha_{21} = \frac{a_1}{a_2} \tag{4} \end{eqnarray} \]

The model above is the Lotka-Volterra model of competition for two species. Notice that it has a distinctive resemblance to previous models of population growth we studied earlier in the course.

Q3. If Consumer 2 is absent, what does the equation for \(\frac{dN_1}{dt}\) look like? What kind of population growth is this? What will be the equilibrium population size of Consumer 1 in this scenario? How would you interpret the constant \(K_1\)?

Q4. Assume that in addition to Consumer 2 being absent, the current population of Consumer 1 is very small. In that case, the ratio \(\frac{N_1}{K_1}\) is much smaller than 1, which is to say, negligible. That being the case, what kind of population growth will Consumer 1 experience? How would you interpret the constant \(r_1\)?

Notice that the constants \(\alpha_{12}\) and \(\alpha_{21}\) are tied to each other via the relationship \(\alpha_{12} \alpha_{21}= 1\). This turns out to be critical for the coexistence prospects of species 1 and 2, as we will see in the exercises below.

The code below defines our Lotka-Volterra model and the plotting function:

LV_Model = 
  function(
    initial_N,
    intrinsic_growth_rate,
    carrying_capacity,
    competition_coefficient,
    final_time,
    time_step
  ){
      LV = 
        function(t, state, parameters){
           with(as.list(c(state, parameters)), {
             N = c(N1, N2)
             dNdt = as.numeric(r * N * (1 - A %*% N / K))
             list(c(dNdt[1], dNdt[2]))
           })
          }
      
      times = seq(0, final_time, by = time_step)
      
      parameters = 
        list(
          r = intrinsic_growth_rate,
          K = carrying_capacity,
          A = 
            matrix(
              c(1, competition_coefficient[1], competition_coefficient[2], 1), 
              byrow = TRUE, 
              nrow = 2
            )
        )
      
      state = c(N1 = initial_N[1], N2 = initial_N[2])
      
      out = ode(y = state, times = times, func = LV, parms = parameters)
      
      return(
        list(
          parameters = 
            list(
              intrinsic_growth_rate, 
              carrying_capacity, 
              competition_coefficient
            ),
          initial_conditions = initial_N,
          state = out
        )
      )
   }

Plot_LV = 
  function(model){
    as.data.frame(model$state) |>
    pivot_longer(-time, names_to = 'species', values_to = 'N') |>
    ggplot(aes(time, N, group = species, color = species)) +
    geom_line(size = 1) +
    expand_limits(y = 0) +
    labs(
      x = 'Time',
      y = 'Abundance'
    )
  }

Q5. Consider two plant species with the same intrinsic growth rates \(r_1 = r_2 = 1\) and carrying capacities \(K_1 = K_2 = 50\), and suppose that \(\alpha_{12} = 1.01\) and \(\alpha_{21} = \frac{1}{1.01}\). Run the Lotka-Volterra competition model by plugging these parameter values in the calling code below. Do both species coexist in the end? If not, which species wins?

# call the Lotka-Volterra model 
model = 
  LV_Model(
    initial_N = c(30, 30), 
    intrinsic_growth_rate = c(r1, r2), 
    carrying_capacity = c(K1, K2), 
    competition_coefficient = c(alpha_12, alpha_21), 
    final_time = 1000, 
    time_step = 1
  )

# plot results
Plot_LV(model)

Q6. Reverse the values of the constants \(\alpha_{12} = \frac{1}{1.01}\) and \(\alpha_{21} = 1.01\) and rerun the model. Who wins now?

It might seem that we could try to tune the other parameters – intrinsic growth rates and carrying capacities – to find a scenario where neither species wins and both coexist. However, that is impossible given the forced relationship between \(\alpha_{12}\) and \(\alpha_{21}\). (More on this below.)

Q7. Let’s pretend for a second that we can choose \(\alpha_{12}\) and \(\alpha_{21}\) independently. Run the model with \(\alpha_{12} = 0.3\), \(\alpha_{21} = 0.6\). Do both species coexist? If not, who wins?

Clearly, the constants \(\alpha_{12}\) and \(\alpha_{21}\) are important in determining the outcome of competition between species 1 and 2. So if we are interested in the outcome of competition, we want to understand the biological meaning of these constants.

Let’s focus on the growth of species 1 for a minute. If species 2 is absent, the population of species 1 follows logistic growth, \(\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{K_1}\right)\). The term in parenthesis, \(1-\frac{N_1}{K_1}\), slows down the growth of the population as \(N_1\) approaches the carrying capacity \(K_1\). When we add species 2, it also contributes to the regulation of population growth of species 1: \(\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1 + \alpha_{12}N_2}{K_1}\right)\). Now the relevant ratio is not \(\frac{N_1}{K_1}\), but \(\frac{N_1 + \alpha_{12}N_2}{K_1}\). Now the equilibrium abundance of species 1 is lower than its carrying capacity:

\[ \frac{dN_1}{dt} = 0 \;\;\; \textrm{if} \;\;\; N_1 = K_1 - \alpha_{12}N_2 \;\;\; \textrm{(or if} \;\;\; N_1 = 0 \textrm{)} \tag{5} \]

Similarly for species 2:

\[ \frac{dN_2}{dt} = 0 \;\;\; \textrm{if} \;\;\; N_2 = K_2 - \alpha_{21}N_1 \;\;\; \textrm{(or if} \;\;\; N_2 = 0 \textrm{)} \tag{6} \]

What is the ecological meaning of \(\alpha_{12}\) and \(\alpha_{21}\)?

Looking again at the regulation term \(1-\frac{N_1 + \alpha_{12}N_2}{K_1}\), we see that \(\alpha_{12}\) quantifies the degree to which species 2 slows down the growth of species 1 relative to how much species 1 slows itself down. While each additional individual of species 1 slows down the growth rate of the population by \(\frac{r_1}{K_1}\), each additional individual of species 2 slows it down by \(\alpha_{12}\frac{r_1}{K_1}\). In other words:

\[ (\textrm{competititve impact of sp 2 on sp 1}) = \alpha_{12} \times (\textrm{competititve impact of sp 1 on itself}) \] Similarly,

\[ (\textrm{competititve impact of sp 1 on sp 2}) = \alpha_{21} \times (\textrm{competititve impact of sp 2 on itself}) \] Given their roles in quantifying the mutual competitive impacts, these constants are called competition coefficients. In ecological jargon, we would say that

The competition coefficient \(\alpha_{ij}\) is the degree to which species \(j\) regulates species \(i\) relative to the degree to which species \(i\) regulates itself.

Notice that if \(\alpha_{12} < 1\), then species 1 is more strongly regulated by itself than by species 2. On the other hand if \(\alpha_{12} > 1\), then species 1 is more strongly regulated by species 2 than by itself. In other words,

\[ \begin{eqnarray} \alpha_{12} < 1 &\Rightarrow& \textrm{species 1 is species 1's worst enemy} \\ \alpha_{12} > 1 &\Rightarrow& \textrm{species 2 is species 1's worst enemy} \end{eqnarray} \]

Q8. Which species regulates species 2 more strongly when \(\alpha_{21} < 1\)? Which species regulates species 2 more strongly when \(\alpha_{21} > 1\)?

Coexistence

One of the most striking characteristics of nature is its biodiversity. For example, the lowland rainforests of Borneo are home to 10,000 species of plants, more than the entire continent of Africa, while occupying 1/40th of its land area.

Figure: Plant biodiversity in Changmai, Thailand. Photo by Mandy Choi

Most people feel that the world’s biodiversity is one of nature’s gifts to be cherished and preserved. But in order to conserve it we must understand it. In fact, one of the central questions in community ecology is how to make sense of this opulence. All plants in a forest compete for the same set of resources. But if there is competition, how come there is no winner? How can so many species coexist in the same location?

Figure: Researcher surveying biodiversity in Borneo, one of the world’s richest forests. Photo from heartofborneo.org

Staying true to the KISS principle in science, we will try to gain insight into this big and important question using our simple Lotka-Volterra model. For if we can’t understand coexistence of two species in a model, we have little chance of explaining coexistence of tens of thousands of species in nature.

So let’s ask ourselves:

When will the two species be able to coexist?

To answer this question we must look in the model and see what needs to be true for both species to have positive abundances in equilibrium.

Looking again at the equilibrium conditions for species 1 (Equation 5) and species 2 (Equation 6), we can write the equilibrium abundances \(N_1\) and \(N_2\) in terms of the constants:

\[ \begin{eqnarray} (1-\alpha_{12}\alpha_{21}) \, N_1 = K_1 - \alpha_{12}K_2 \\ (1-\alpha_{12}\alpha_{21}) \, N_2 = K_2 - \alpha_{21}K_1 \tag{7} \end{eqnarray} \]

Looks like we can divide by \(1-\alpha_{12}\alpha_{21}\) and write \(N_1=\frac{K_1-\alpha_{12}}{1-\alpha_{12}\alpha_{21}}\) and \(N_1=\frac{K_2-\alpha_{21}}{1-\alpha_{12}\alpha_{21}}\). However, we run into a problem: \(\alpha_{12}\alpha_{21} = 1\), which means we can’t divide by \(1-\alpha_{12}\alpha_{21}\) because that would be division by zero. Rather, the only way for the equations above to hold is if \(K_1 = \alpha_{12}K_2\) (or equivalently if \(K_2 = \alpha_{21}K_1\), which is the same thing since \(\alpha_{21} = 1/\alpha_{12}\)). In words: the only way species 1 and 2 could coexist when \(\alpha_{12}\alpha_{21} = 1\) is if their carrying capacities were related by a factor of exactly \(\alpha_{12}\).

Let’s see what happens when this condition is met. Suppose \(r_1 = 1\), \(r_2=1\), \(K_2=30\), and \(\alpha_{12}=1.1\), \(\alpha_{21}=\frac{1}{1.1}\). In words, both species have the same intrinsic growth rate, but species 2 is a stronger competitor than species 1 (since the competitive effect of 2 on 1 is greater than 2 on itself, whereas the competitive effect of 1 on 2 is less than 1 on itself). The coexistence condition above requires that \(K_1=\alpha_{12}K_2=33\). This makes sense: it is saying that species 1 must have a higher carrying capacity to exactly compensate for the competitive advantage of species 2. Let’s feed all of that into our LV model and plot the result:

# parameter values
initial_N1 = 5
initial_N2 = 10
r1 = 1
r2 = 1
K2 = 30
alpha_12 = 1.1
alpha_21 = 1/alpha_12
K1 = alpha_12 * K2

# call the Lotka-Volterra model 
model = 
  LV_Model(
    initial_N = c(initial_N1, initial_N2), 
    intrinsic_growth_rate = c(r1, r2), 
    carrying_capacity = c(K1, K2), 
    competition_coefficient = c(alpha_12, alpha_21), 
    final_time = 50, 
    time_step = .1
  )

N1_final = model$state[nrow(model$state), 2]
N2_final = model$state[nrow(model$state), 3]

# plot results
Plot_LV(model) +
  ggtitle(paste('Final abundances: N1 =', round(N1_final), '     N2 =', round(N2_final)))

How to describe what we see above? It looks like both species coexist and the equilibrium state is \((N_1, N_2) = (10, 21)\). But now let’s check what happens when we change the initial condition.

Q9. Rerun the code above, this time changing the initial abundances to initial_N1 = 7, initial_N2 = 5. Are the final abundances the same as in the example above when we started with different initial abundances?

This reveals an interesting property of the coexistence scenario in this model, which we have not previously seen in the course: the equilibrium is neither stable nor unstable. It is neutral.

A dynamical system is attracted to a stable equilibrium, like a marble resting at the bottom of a well. If the equilibrium is stable, then even if we start with different initial abundances, we’ll end at the same point. In contrast, the system is repelled away from an unstable equilibrium, like a marble balancing precariously on the top of a hill. Unlike both these scenarios, a neutral equilibrium does not attract nor repel the system. If you dislodge it from that state, it will just go rest a certain distance from it at a different neutral equilibrium. Like an object sitting still on a flat surface.

Figure: Illustration of different types of equilibrium. From L. Zhang, PhD thesis (2016)

In our course, we have seen examples of stable and unstable equilibria. Under logistic growth, \(N=0\) is an unstable equilibrium and the carrying capacity \(K\) is a stable equilibrium. As long we don’t start from \(N=0\), eventually \(N\) will reach \(K\) and stay there. The coexistence of two consumers on a single resource is our first example of a neutral equilibrium in ecology.

The name neutral is even more apt when we think about what needs to be true of the two consumers such that they can coexist like this. Looking back at how these constants relate to the parameters of the original consumer-resource model (Equation 4), we see that \(K_1 = \alpha_{12} K_2\) means that both consumers need to have the same growth-to-death ratio, \(\frac{b_1}{d_1} = \frac{b_2}{d_2}\).

One way to frame this finding is that both species need to be equivalent in terms of their interaction with the resource. This bears emphasizing:

The only way two species can coexist on a single resource is neutrally, by being ecologically equivalent.

We will return to this notion of ecological equivalence, also called neutrality, in competitive systems. But for now we should note that this fine-tuning is unlikely to happen in nature by chance. We conclude we shouldn’t expect coexistence based on this model.

Mathematical Note

The fact that Consumers 1 and 2 cannot both have positive equilibrium abundances while feeding on a single resource is very general and holds irrespective of the dynamics of the resource. Equation (2) can be solved exactly for the ratio of the populations of the two consumers. The result is \[ \frac{N_1^{b_2}}{N_2^{b_1}} = k \, e^{(b_1d_2-b_2d_1)t} \] In words, the ratio of the two populations either grows exponentially to infinity if \(b_1d_2 > b_2 d_1\) – signifying that the population of Consumer 2 collapses – or shrinks exponentially to 0 if \(b_1d_2 < b_2 d_1\) – signifying that Consumer 1 collapses. The only other possibility is the fine-tuned scenario \(b_1d_2 = b_2 d_1\), where the ratio is constant in time, meaning both populations grow together or shrink together.

Recall from Q5 that we were, in fact, able to get coexistence in this model without fine-tuning. However, we had to cheat and relax the constraint \(\alpha_{12}\alpha_{21} = 1\). So the question is, can we relax this constraint without cheating?

The answer is Yes, and the way to do it is to add another resource:

\[ \begin{eqnarray} \frac{dR_1}{dt} &=& c_1R_1\left[\left(1-\frac{R_1}{k_1}\right) - a_{11}N_1-a_{12}N_2\right] \\ \frac{dR_2}{dt} &=& c_2R_2\left[\left(1-\frac{R_2}{k_2}\right) - a_{21}N_1-a_{22}N_2\right] \\ \frac{dN_1}{dt} &=& (b_{11}R_1 + b_{21} R_2-d_1)N_1\\ \frac{dN_2}{dt} &=& (b_{12}R_1 + b_{22} R_2-d_2)N_2 \tag{8} \end{eqnarray} \] This model is the logical extension of our 2-consumers-1-resource model, Equation (1), except now we have to use two indices in the consumption and conversion rates \(a_{ij}\) and \(b_{ij}\) to signify consumer \(j\)’s relationship to resource \(i\).

Notice the model assumes that both consumers exploit both resources, albeit to different degrees. In other words, the consumers are generalists, but may differ in their preferences for the different resources.

Q10. Give an example of a two-consumer-two-resource system where both consumers exploit both resources.

Q11. Suppose \(b_{11} > b_{21}\) and \(b_{12} > b_{22}\). In this situation, what is the preferred resource of consumer 1? What is the preferred resource of consumer 2?

Q12. Between a hypothetical scenario where both consumers prefer the same resource and an alternative scenario where each consumer prefers a different resource, which one would be more likely to allow coexistence of the consumers? Explain your answer.

As before, here if we assume faster resource dynamics relative to consumer dynamics, we can simplify the model by getting rid of \(R_1\) and \(R_2\), and rewriting the model in terms of the consumers alone, \(N_1\) and \(N_2\). As it turns out, we get the exact same Lotka-Volterra competition equations (Equation 3)! The crucial difference is how the intrinsic growth rates (\(r_1\), \(r_2\)), carrying capacities (\(K_1\), \(K_2\)), and competition coefficients (\(\alpha_{12}\), \(\alpha_{21}\)) relate to the parameters of the consumer-resource model \(a_{ij}, \; b_{ij}, \; d_i, \; k_i\). The relationship between the parameters, shown below, is not simple enough to be particularly illuminating, so we won’t worry about it here (you can check out Chapter 8 of the textbook for the derivation, but watch out for typos there). However, the important point is that with two resources it is no longer necessarily the case that \(\alpha_{12}\alpha_{21} = 1\). As such, we may proceed with investigating the coexistence conditions, Equation (7).

\[ \begin{eqnarray} r_1 = b_{11} k_1 + b_{21}k_2 - d_1 & \hspace{1cm} & K_1 = \frac{r_1}{a_{11}b_{11}k_1 + a_{21}b_{21}k_2} & \hspace{1cm} \alpha_{12} = \frac{a_{12}b_{11}k_1 + a_{22}b_{21}k_2}{a_{11}b_{11}k_1 + a_{21}b_{21}k_2} \\ r_2 = b_{22} k_2 + b_{12}k_1 - d_2 & \hspace{1cm} & K_2 = \frac{r_2}{a_{22}b_{22}k_2 + a_{12}b_{12}k_1} & \hspace{1cm} \alpha_{21} = \frac{a_{21}b_{22}k_2 + a_{11}b_{12}k_1}{a_{22}b_{22}k_2 + a_{12}b_{12}k_1} \tag{9} \end{eqnarray} \] We can rewrite Equation (7) to find the equilibrium population sizes of species 1 and 2:

\[ \begin{eqnarray} N_1 = \frac{K_1 - \alpha_{12}K_2}{1-\alpha_{12}\alpha_{21}} \\ N_2 = \frac{K_2 - \alpha_{21}K_1}{1-\alpha_{12}\alpha_{21}} \tag{10} \end{eqnarray} \]

Coexistence will be possible when both \(N_1\) and \(N_2\) are positive. This can only happen under two different scenarios: either the denominator and both numerators are positive (Scenario 1), or the denominator and both numerators are negative (Scenario 2). Mathematically, these scenarios are characterized as follows:

\[ \begin{eqnarray} \textrm{Scenario 1:} \hspace{1cm} \alpha_{12}\alpha_{21} < 1 \hspace{1cm} \textrm{and} \hspace{1cm} K_1 > \alpha_{12} K_2 \hspace{1cm} \textrm{and} \hspace{1cm} K_2 > \alpha_{21} K_1 \\ \\ \textrm{Scenario 2:} \hspace{1cm} \alpha_{12}\alpha_{21} > 1 \hspace{1cm} \textrm{and} \hspace{1cm} K_1 < \alpha_{12} K_2 \hspace{1cm} \textrm{and} \hspace{1cm} K_2 < \alpha_{21} K_1 \tag{11} \end{eqnarray} \] Under any other circumstances, either species 1 wins and species 2 goes extinct, or vice-versa. However, even between those two scenarios, only one of them will reliably lead to coexistence, because in that scneario coexistence is a stable equilibrium (see explanation above). In the other one, coexistence is unstable, and therefore either species 1 or species 2 will win, depending on initial abundances.

To simplify things, let’s assume both species have the same carrying capacity (none of what follows depends on this assumption, but it does make things easier to see). In that case, we can write the two scenarios as:

\[ \begin{eqnarray} \textrm{Scenario 1:} \hspace{1cm} \alpha_{12} < 1 \hspace{1cm} \textrm{and} \hspace{1cm} \alpha_{21} < 1 \\ \\ \textrm{Scenario 2:} \hspace{1cm} \alpha_{12} > 1 \hspace{1cm} \textrm{and} \hspace{1cm} \alpha_{21} > 1 \end{eqnarray} \]

These scenarios are easy to interpret biologically. In one of them, each species has a stronger competitive impact on itself than on the other species. In the other scenario, the reverse is true: each species impacts the other more than it impacts itself. In other words, in one of the scenarios, intraspecific competition \(>\) interspecific competition, and in the other the reverse is true.

Q13. In which scenario (Scenario 1 or Scenario 2) is intraspecific competition \(>\) interspecific competition?

Q14. Use the code below to find out which scenario (1 or 2) gives stable coexistence. For each scenario, do steps (a) and (b) as follows: a) Choose values for \(\alpha_{12}\) and \(\alpha_{21}\) that satisfy the conditions of the scenario. The code will then calculate the predicted equilibrium abundances of the two species, N1_equil and N2_equil. Use those abundances as the initial abundances (replace the ??) and verify that those abundances do not change over time – which confirms that those abundances form an equilibrium. Show your plot. b) Run the model again, this time with different initial abundances, and describe what happens. Show your plot. Make sure to do steps (a) and (b) for both scenarios 1 and 2.

## set parameter values
r = 0.1
K1 = 100
K2 = 100
alpha12 = ??
alpha21 = ??

## calculate predicted equilibrium abundances
N1_equil = (K1 - alpha12 * K2) / (1 - alpha12 * alpha21)
N2_equil = (K2 - alpha21 * K1) / (1 - alpha12 * alpha21)

## call the Lotka-Volterra model 
model = 
  LV_Model(
    initial_N = c(??, ??), 
    intrinsic_growth_rate = c(r, r), 
    carrying_capacity = c(K1, K2), 
    competition_coefficient = c(alpha12, alpha21), 
    final_time = 1000, 
    time_step = 1
  )

## plot results, adding horizontal dashed lines marking the predicted equilibrium abundances
Plot_LV(model) +
  ggtitle(paste('Predicted equilibrium abundances: N1 =', round(N1_equil), '     N2 =', round(N2_equil)))

The scenario where which species wins depends on initial conditions is called priority effects, in reference to how the winner is the species that happens to become abundant before the other species invades. By contrast, the scenario where species coexist regardless of initial abundances is caleld stable coexistence.

Q15. Having gone through all the steps for both scenarios in Q14, explain which scenario above has priority effects and which constitutes stable coexistence.

Structural stability

Notice the difference for coexistence between having a single resource and two resources. In the single-resource case, the only way the two consumers could coexist is if their carrying capacities satisfied a precise relationship that implied the two species have the exact same impact on the resource. With two species, by contrast, there is a range of carrying capacities that satisfy the coexistence conditions in Equation (11). This contrast between there being a range of possible parameters leading to coexistence under two resources and a single set of parameter values leading to coexistence under a single resource reflects another type of stability that the 2-resource scenario has which the 1-resource scenario lacks: We say coexistence between two consumers in the 2-resource model is structurally stable, while in the single-resource scenario coexistence is structurally unstable. Structural stability is important in the real world because only structurally stable coexistence can withstand changes to its parameters and still support coexistence. These parameter changes can result from changes in environmental conditions such as climate change or introduction of a new species. The more structurally stable the coexistence scenario, the more change it can absorb without collapsing.

Notice the difference between structural stability and the other kind of stability which we have been focusing on so far, whose full name is dynamical stability. Dynamically stable coexistence occurs when both species have positive equilibrium abundances and will reach those abundances almost regardless of initial abundances (i.e. they are attracted to that equilibrium, like a marble at the bottom of a well). Structurally stable coexistence means that coexistence will still be dynamically stable even if we change some properties of the system, like species carrying capacities (or the shape of the well in the marble example).

You saw another instance of structural stability analysis when we discussed Robert May’s work on food webs in Module 6, Predator-Prey Dynamics. Recall that May showed that food webs with a large number of species and/or strong interactions between species were less likely to have a dynamically stable equilibrium than food webs with a small number of species and/or weak interactions. In other words, May showed that large food webs are less structurally stable than small ones. This raises the question Why are there large food webs in nature? The answer, as we have seen in Module 6 relates to mechanisms that weaken species interactions and create specific patterns of interactions (motifs) within the full web. The fact that large food webs are unlikely to be dynamically stable without such mechanisms reflects their low structural stability.

Bonus Q (1 extra point): Based on the discussion in the box above, describe structural stability in your own words, and explain its relevance for conservation biology in a world undergoing climate change (If you choose to look this up, make sure to cite your source).

The relationship between the Lotka-Volterra competition coefficients \(\alpha_{12}\), \(\alpha_{21}\) and the consumer-resource model, as shown in Equation (9), is

\[ \begin{eqnarray} \alpha_{12} = \frac{a_{12}b_{11}k_1 + a_{22}b_{21}k_2}{a_{11}b_{11}k_1 + a_{21}b_{21}k_2} \\ \\ \alpha_{21} = \frac{a_{21}b_{22}k_2 + a_{11}b_{12}k_1}{a_{22}b_{22}k_2 + a_{12}b_{12}k_1} \end{eqnarray} \]

Q16. Consider a case where \(a_{12} < a_{11}\) and \(a_{22} < a_{21}\). Recalling that \(a_{ij}\) is the consumption rate of resource \(i\) by consumer \(j\), which species consumes more of resource 1, and which species consumes more of resource 2?.

Q17. Still considering the situation where \(a_{12} < a_{11}\) and \(a_{22} < a_{21}\), notice that this means \(\alpha_{12} < 1\). Recalling what was said above, what is the biological meaning of this inequality, in terms of competitive impacts of one species on another?

Q18. Again considering the situation where \(a_{12} < a_{11}\) and \(a_{22} < a_{21}\), what can we say about \(\alpha_{21}\), and the corresponding biological meaning?

Q19. Given your answers to Q16 and Q17, what do you deduce about coexistence between these two species? Connecting this conclusion to your answer to Q15, what general rule do you arrive at regarding what needs to be true about the preferences/affinities of the two consumers for each of the two resources if the two species are to coexist?

Testing the Lotka-Volterra model

The Lotka-Volterra competition equations are the foundation of competition theory and coexistence theory. Here, we will look at an experimental study (Vandermeer 1969, Ecology 50:3) which sought to test the Lotka-Volterra model (abbreviated LV from now on) by measuring the LV parameters and verifying whether competition outcomes matched the model predictions.

Methodological Note

Before proceeding, we need to know how to measure the LV constants. As you recall, in order to make predictions about coexistence of two species, we must know their carrying capacities \(K_1\), \(K_2\), and the competition coefficients \(\alpha_{12}\), \(\alpha_{21}\). Measuring the \(K\)’s is straightforward: we just grow the species separately and measure their asymptotic (i.e. average long-term) population size. But the competition coefficients are trickier. Looking back at Equation (3) defining our model, we see that if we can grow the species together and measure their growth rates \(\frac{dN_1}{dt}\) and \(\frac{dN_2}{dt}\) as their abundances change over time, then our empirical estimate for \(\alpha_{12}\) will be the slope of the following linear regression: \[ \begin{eqnarray} \frac{K_1}{r_1}\frac{1}{N_1}\frac{dN_1}{dt} + N_1 = K_1 - \alpha_{12}N_2 \end{eqnarray} \]

Analogously, our empirical estimate for \(\alpha_{21}\) will be the slope of the regression:

\[ \begin{eqnarray} \frac{K_2}{r_2}\frac{1}{N_2}\frac{dN_2}{dt} + N_2 = K_2 - \alpha_{21}N_1 \end{eqnarray} \]

Finally, notice that the competition coefficients can be arranged in a matrix, called the community matrix: \[ \pmatrix{1 & \alpha_{12} \\ \alpha_{21} & 1} \]

where the matrix entry on the \(i\)-th row and \(j\)-th column represents the competitive impact of species \(j\) on species \(i\) relative to the impact of species \(i\) on itself (and therefore the diagonal elements are 1 by definition). The community matrix offers at a glance a sense of how each species in the community interacts with all others. This seems unnecessarily complicated when we only have 2 species, but becomes handy when we have more species, as in the Vandermeer 1969 study, to which we now turn.

The study consisted of growing different species of paramecium – aquatic unicellular organisms that eat bacteria and dead organic matter – in isolation and in combination. Upon growing Paramecium aurelia, Paramecium bursaria, and Paramecium caudatum in isolation in test tubes, Vandermeer ascertained that all three grew logistically, and reached the following carrying capacities (in cells per 0.5 ml of culture):

species	K
P. aurelia	671
P. bursaria	230
P. caudatum	366

Figure: P. aurelia by Barfooz. P. bursaria by Picturepest. P. caudatum by Deuterostome

Upon growing each of the 3 possible pairs of species together, Vandermeer estimated the following community matrix:

	PA	PB	PC
PA	1.0	-2.0	1.75
PB	0.5	1.0	0.85
PC	0.3	0.5	1.00

We can now use those two tables to predict the competition outcomes of PA growing together with PB, PA with PC, and PB with PC.

Q20. What is the biological meaning of the negative coefficient \(\alpha_{AB} = -2\)? Hint: Look again at Equation (3) and think about what happens to the growth rate of species 1 when \(N_2\) increases if \(\alpha_{12}\) is negative.

Example: PA with PC. We have \(K_A = 671\), \(K_C = 366\), \(\alpha_{AC} = 1.75\), and \(\alpha_{CA} = 0.3\). Based on the competition coefficients, we have \(\alpha_{AC}\alpha_{CA} <1\). Also, \(K_A > \alpha_{AC}K_C\) since \(671 > 1.75 \times 366\) and \(K_C > \alpha_{CA}K_A\) since \(366 > 0.3 \times 671\). We conclude that this corresponds to Scenario 1 in Equation (11), and therefore PA and PC should stably coexist.

How does our prediction compare with the result of the experiment? Here is Figure 10 from the paper:

From Vandermeer 1969.

The LV predictions match observations.

Q21. Predict the competition outcome between PB and PC. Compare your prediction to the observations shown in the Fig below, and discuss.

From Vandermeer 1969.

While this study was consistent with the predictions of the LV model, we must keep in mind that the model is simplistic. For example,

• It assumes that the competitive impact of a species on another always scales linearly (eg doubling the abundance of species 2 always doubles its competitive impact on species 1), which is not realistic

• It ignores the influence of other important determinants of species coexistence, such as immigration and stochastic events.

Indeed, many subsequent studies failed to confirm the predictions of the LV model in experimental and observational settings (eg Neill 1974, Seifert and Seifert 1976).

Its simplism notwithstanding, the LV model is of enormous importance to community ecology, as it codified previously vague notions of competitive coexistence into a formal set of principles. Just as Verhulst’s logistic growth is the simplest model of self-regulated growth, which we now can interpret as resulting from intraspecific competition, Lotka and Volterra’s equations form the simplest model of interspecific competition, and have been foundational for much of the subsequent modeling work.

In summary, we have learned the following from the Lotka-Volterra competition model

• Two-species coexistence occurs when each species regulates itself more than the other species. This can only happen if there is more than one resource. With a single resource, whichever species is better at consuming the resource wins. With at least two resources, each species can survive by being the better competitor for a different resource.

• Coexistence requires an opposition between a species’ frequency in the community (i.e. its relative abundance) and its fitness as measured by the species growth rate: the higher the frequency, the lower the fitness, and vice-versa. This phenomenon is called negative frequency dependence.

• Priority effects occur when each species regulates the other species more than itself. This creates a positive feedback cycle, where a species fitness increases as its relative abundance increases (since it gets less and less competition from its worst enemy, ie the other species). In this case, coexistence is not possible, and the winner depends on initial conditions.

The latter point above is the foundation of the concept of niche differentiation in community ecology. In Part II we will look at what some of this later work had to say about the phenotypes of species that are able to coexist.

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